Good question. Haven't actually read it. Do they mean constant for large thicknesses, or just constant at a given location, meaning that the strength is assumed constant irrespective of changes in the stress state or PWP at that location? Seems to me if Su varies with depth (according to CPT, VST, or whatever), you should treat it that way (by layers in the model). If you use a strength typical of the upper part of the deposit and it actually increases with depth, you would be throwing away some usable resistance.
I had Duncan reviewing a project of mine a few years ago. Since we couldn't actually do complete SHANSEP calculations with Slope/w, we faked it with a phi-cu calculation (which gives undrained shear strength as a function of effective normal stress on the failure surface), which he was willing to accept, since we explained our rationale and tweaks. We lucked out, because you can't always do that - see CC Ladd's 1986 Terzaghi lecture for explanation of why phi-cu goes bad with some geometries, like steep exit of slide surface. (At least I think that's where I saw it. I'm home now and don't keep duplicate references.) When we used FLAC, we were able to program in the SHANSEP-type strength ratio and apply it to sigma'1c (pre-earthquake major principal stress). Unfortunately the strength ratio or the phi-cu you fake it with varies with stress path.
Let me know if you find an answer to the original question.